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Theorem curry2f 6214
Description: Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
curry2.1  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
Assertion
Ref Expression
curry2f  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G : A
--> D )

Proof of Theorem curry2f
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fovrn 5990 . . . . 5  |-  ( ( F : ( A  X.  B ) --> D  /\  x  e.  A  /\  C  e.  B
)  ->  ( x F C )  e.  D
)
213com23 1157 . . . 4  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B  /\  x  e.  A
)  ->  ( x F C )  e.  D
)
323expa 1151 . . 3  |-  ( ( ( F : ( A  X.  B ) --> D  /\  C  e.  B )  /\  x  e.  A )  ->  (
x F C )  e.  D )
4 eqid 2283 . . 3  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
53, 4fmptd 5684 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( x  e.  A  |->  ( x F C ) ) : A --> D )
6 ffn 5389 . . . 4  |-  ( F : ( A  X.  B ) --> D  ->  F  Fn  ( A  X.  B ) )
7 curry2.1 . . . . 5  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
87curry2 6213 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
96, 8sylan 457 . . 3  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
109feq1d 5379 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( G : A --> D  <->  ( x  e.  A  |->  ( x F C ) ) : A --> D ) )
115, 10mpbird 223 1  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G : A
--> D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251  (class class class)co 5858   1stc1st 6120
This theorem is referenced by:  curry2ima  23247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123
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